
Collisions of bubble walls and shocks in the plasma. GW's produced from this process depends only on the dynamics of the scalar field.
The “envelope approximation” is used in the numerical simulations to estimate the GW spectrum, given by [1]}
(analytical calculations of the GW spectrum from this process can be found in Ref. [2]})
\(h^{2} \Omega _{\mathrm {\rm col}}(f)=1.67 \times 10^{-5}\left(\frac{H_{*}}{\beta }\right)^{2}\left(\frac{\kappa _{\rm col}\alpha }{1+\alpha }\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}}\left(\frac{0.11 v_{w}^{3}}{0.42+v_{w}^{2}}\right) S_{\mathrm {\rm col}}(f).\) 

Sound waves in the plasma generated subsequently after the bubble collisions. Numerical
simulations indicate that the durations of sound waves and turbulence as active sources of GW's are typically much longer than the collisions of the bubble walls. This process contributes a GW spectrum desribed by [3]}
\(h^{2} \Omega _{\mathrm {sw}}(f)=2.65 \times 10^{-6}\left(\frac{H_{*}}{\beta }\right)\left(\frac{\kappa _{\rm sw} \alpha }{1+\alpha }\right)^{2}\left(\frac{100}{g_{*}}\right)^{\frac{1}{3}} v_{w} S_{\mathrm {sw}}(f).~~~~~~~~~~~~~~~\) 

Turbulence in the plasma formation after the bubble collisions. Simulations show that only a small fraction \(\epsilon \sim 5-10\%\)  of the bulk motion from the bubble walls is converted into turbulence. However, GW's produced from this process could play a dominant role at high frequencies, as the GW signals from sound waves decay much faster. The GW spectrum from turbulence can be parameterized as [4]}
\(h^{2} \Omega _{\mathrm {turb}}(f)=3.35 \times 10^{-4}\left(\frac{H_{*}}{\beta }\right)\left(\frac{\kappa _{\mathrm {turb}} \alpha }{1+\alpha }\right)^{\frac{3}{2}}\left(\frac{100}{g_{*}}\right)^{1 / 3} v_{w} S_{\mathrm {turb}}(f).~~~~~~~~~\) 

